Syllogistic - University of Washington

[Pages:9]Aristotle's Syllogistic

1. Propositions consist of two terms (a subject and a predicate) and an indicator of quantity/quality: `every', `no', `some', `not every'.

TYPE

TAG

IDIOMATIC

TECHNICAL

ABBREVIATION

Universal

a

Affirmative

Every F is G

G belongs to every F

GaF

G is predicated of every F

F is in G as in a whole

Universal

e

Negative

No F is G

G belongs to no F

GeF

G is predicated of no F

Particular

i

Affirmative

Some F is G

G belongs to some F

GiF

G is predicated of some F

G is related to F in part

Particular

o

Negative

Some F is not G

G does not belong to some F

GoF

G belongs to not every F

2. Contraries and Contradictories (cf. De Int. 17bl6-28)

a. p and q are contradictories iff

i. p and q cannot both be true, and ii. necessarily, either p or q is true.

a and o are contradictories e and i are contradictories

b. p and q are contraries iff

i. p and q cannot both be true, and ii. p and q can both be false.

a and e are contraries

c. p and q are subcontraries iff they are the contradictories of a pair of contraries. That is, p and q are subcontraries iff

i. p and q cannot both be false, and ii. p and q can both be true

i and o are subcontraries

3. Syllogistic Figures

? `P' represents the predicate of the conclusion (major term), and `S' represents the subject of the conclusion (minor term). Aristotle calls these terms the "extremes."

? `M' represents the term that is common to the premises but absent from the conclusion. Aristotle calls this the "middle term."

? Aristotle calls the premise containing `P' the "major premise," and the premise containing `S' the "minor premise."

? `*' represents any of the four quantity/quality indicators (`a', `e', `i', `o').

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First Figure

Second Figure

Third Figure

P*M

M*P

P*M

M*S

M*S

S*M

P*S

P*S

P*S

4. Syllogistic Moods

When we replace each `*' with a quantity/quality indicator we get a triple of propositions, . Each distinct triple (e.g., , , etc.) constitutes a mood. (Aristotle, however, did not call these `moods'; this expression is due to later commentators.)

There are 64 such moods (43 = 64). Each mood can occur in each of the three figures. Hence, there are 3 x 64 = 192 candidate syllogisms among the various mood/figures.

Each candidate syllogism can be labeled as to mood and figure as in the following example:

GaF

GeH

The mood is , and the figure is the second; we can

FeH

abbreviate this as -2

Aristotle attempts to identify and prove the validity of the syllogisms ( = valid mood/figures) and to identify and show the invalidity of the others. [More precisely, he attempts to show which premise-pairs do, and which do not, yield syllogistic conclusions; to prove the validity of those that do; and to show (by counterexample) the invalidity of those that do not.]

5. The Valid Moods

Aristotle recognizes the following as syllogisms (i.e., as valid moods):

First Figure

Second Figure

Third Figure

(Barbara)

(Cesare)

(Darapti)

(Celarent)

(Camestres)

(Felapton)

(Darii)

(Festino)

(Disamis)

(Ferio)

(Baroco)

(Datisi)

(Bocardo)

(Ferison)

The standard predicate-logic representations of all but two of these are valid; but Darapti and Felapton

require existential import (of the middle term). If the minor term also has existential import, we get four

additional "weakened" syllogisms with valid predicate-logic representations:

First Figure

Second Figure

(Barbari) (Celaront)

(Cesaro) (Camestros)

Aristotle excludes these from among the valid syllogisms not because they require existential import, but because their premises support stronger (universal) conclusions than the (particular) ones claimed here.

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6. Disproof (rejection) procedure by counterexample Example at 26a36-38:

"Suppose that B belongs to no C, and that A either belongs or does not belong to some B or does not belong to every B; there will be no deduction in this case either. Let the terms be white--horse--swan, and white--horse--raven."

We have been given two premise-pairs of the first figure, and :

Case 1

Premises

AiB

BeC

There are four candidate syllogisms in each case:

Case 2

AoB BeC

Positive

AaC

AaC

Conclusions

AiC

AiC

Negative

AeC

AeC

Conclusions

AoC

AoC

Aristotle gives two sets of values for the variables. The first set has a positive relation between A and C; the second set has a negative relation between A and C.

Positive relation:

A = white B = horse C = swan

Negative relation:

A = white B = horse C = raven

The swan example (positive relation) disproves the four candidates with negative conclusions.

The raven example (negative relation) disproves the four candidates with positive conclusions. So Aristotle in effect gives us counter-examples to all eight candidates. In every case, the premises are true and the conclusion is false, so none of the candidates is a syllogism:

Major premise Minor premise

Positive Conclusions Negative Conclusions

Case 1

Some horse is white

No swan is a horse No raven is a horse

Every raven is white Some raven is white

No swan is white Some swan is not white

For another example see 27a18-21.

Case 2

Some horse is not white

No swan is a horse No raven is a horse

Every raven is white Some raven is white

No swan is white Some swan is not white

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7. Syllogistic as a Natural Deduction System

RULES:

Premise-introduction: Any premise-pair of any of the three figures may be assumed as premises.

Barbara: Celarent: Darii: Ferio:

From AaB and BaC, infer AaC From AeB and BaC, infer AeC From AaB and BiC, infer AiC From AeB and BiC, infer AoC

Simple Conversion e-Conversion From AeB infer BeA i-Conversion From AiB infer BiA

Accidental Conversion a-Conversion From AaB infer BiA

Reduction per impossibile:

r may be inferred from the premise-pair {p, q}, if either the contradictory or the contrary of q has been inferred from the premise-pair {p, the contradictory of r}.

That is, to prove a syllogistic conclusion, show that the assumption of its contradictory (along with one of the premises) leads to the contradictory or the contrary of the other premise.

[Examples of reduction per impossibile: For the contradictory of q inferred from {p, the contradictory of r}, see 28bl7-18. For the contrary of q inferred from {p, the contradictory of r}, see 29a37-39.]

is a syllogism iff it is a mood of one of the three figures and r is deducible from {p, q}

by the rules above.

Aristotle eventually shows how Darii and Ferio can be obtained as derived rules and hence dropped from the set of primitive rules (cf. 29b1-25).

8. Some Proofs of Syllogisms (texts omitted in Fine/Irwin) a. Cesare (27a5-9)

"Let M be predicated of no N but of every X. Then, since the privative converts, N will belong to no M. But M was assumed to belong to every X, so that N belongs to no X (for this has been proved earlier)."

1. MeN 2. MaX 3. NeM 4. NeX

Premise Premise 1 e-Conversion 2,3 Celarent

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b. Camestres (27a9-13)

"Next, if M belongs to every N but to no X, then neither will N belong to any X. For if M belongs to no X, neither does

X belong to any M; but M belonged to every N; therefore, X will belong to no N (for the first figure has again come about). And since the privative converts, neither will N belong to any X, so that there will be the same deduction. (It is also possible to prove these results by leading to an impossibility.)"

1. MaN 2. MeX 3. XeM 4. XeN 5. NeX

Premise Premise 2 e-Conversion 1,3 Celarent 4 e-Conversion

c. Baroco (27a36-b1)

"Next, if M belongs to every N but does not belong to some X, it is necessary for N not to belong to some X. For if it

belongs to every X and M is also predicated of every N, then it is necessary for M to belong to every X: but it was assumed not to belong to some. And if M belongs to every N but not to every X, then there will be a deduction that N does not belong to every X. (The demonstration is the same.)"

1. MaN 2. MoX

3. NaX 4. MaX

5. NoX

Premise Premise Assumption for RPI 1,3 Barbara

1-4 RPI

From 1 and the contradictory of 5 (= 3) we have derived the contradictory of 2 (= 4). Hence we may (by rule RPI) derive 5 from 1 and 2.

d. Darapti (28a18-25)

"When both P and R belong to every S, it results of necessity that P will belong to some R. For since the positive premise converts, S will belong to some R; consequently, since P belongs to every S and S to some R, it is necessary for P to belong to some R (for a deduction through the first figure comes about). It is also possible to carry out the demonstration through an impossibility or through the setting-out (ekthesis). For if both terms belong to every S, then if some one of the Ss is chosen (for instance N), then both P and R will belong to this; consequently, P will belong to some R."

i. By conversion:

1. PaS 2. RaS 3. SiR 4. PiR

Premise Premise 2 a-Conversion 1,3 Darii

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ii. By "ekthesis":

1. PaS

2. RaS

3. Let N be one of the S's 4. P(N) 5. R(N)

6. PiR

Premise Premise Assumption 1,3 Barbara ? 2,3 Barbara ?

? (ekthesis)

What is the syntactical status of `N'? It may be an implicitly quantified class term (so that (4) is, in effect, PaN), or a name of an individual (so that (4) is, in effect, n is P). If it is the former, Aristotle seems to have a problem. For if the implicit quantifier is universal the rule of "ekthesis" would then be: from PaN and RaN, infer PiR. But that has exactly the logical form of Darapti, and so this proof of Darapti would be circular.

So, if `N' is an individual rather than a class term, what is the rule justifying line 6? It appears to be something like Existential Generalization: if there is an individual, x, such that both Fx and Gx, infer FiG).

e. Proofs of Two "Derived Rules" i. Darii (29b9-11)

"The particular deductions in the first figure are brought to completion through themselves, but it is also possible to prove them through the second figure, leading away to an impossibility. For instance, if A belongs to every B and B to some C, we can prove that A belongs to some C. For if it belongs to no C and to every B, then B will not belong to any C (for we know this through the second figure)."

1. AaB 2. BiC 3. AeC 5. BeC 6. AiC

ii. Ferio (29b12-15)

Premise Premise Assumption for RPI 1,3 Cesare (derived rule) 1

1-5 RPI

"If A belongs to no B and B to some C, then A will not belong to some C. For if it belongs to every C but to no B, then neither will B belong to any C (this was the middle figure)."

1. AeB

2. BiC 3. AaC 4. BeC -

5. AoC

Premise Premise Assumption for RPI 1,3 Camestres (derived rule)2

1-4 RPI

1 Aristotle omits the material in angle brackets, neglecting to mention that he is using the conversion rule as well as the "second figure" syllogism (Cesare).

2 Aristotle is appealing to only the first four lines of his proof of Camestres.

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9. Aristotle's Square and the Modern Square

a. Aristotle's square of opposition:

contraries

a

e

implies

implies

i

o

subcontraries

Notice all the logical relationships contained in this diagram: a and o are contradictories (always have opposite truth-values), as are e and i. a logically implies i, and e logically implies o. a and e are contraries (can't both be true, but may both be false), while i and o are subcontraries (can't both be false, but may both be true).

Aristotle's square of opposition is notoriously out of step with the "modern" square. That is, when Aristotle's a, e, i, and o propositions are "translated" in a standard way into the notation of contemporary first-order logic, these traditional relationships of entailment and contrariety do not survive.

b. The modern square:

a

e

i

o

All that remains of the traditional square is that a and o are contradictories, and that e and i are contradictories. That is because the "modern" versions of these propositions look like this:

a All Fs are Gs e No Fs are Gs i Some Fs are Gs o Some Fs are not Gs

x (Fx Gx) x (Fx ?Gx) x (Fx Gx) x (Fx ?Gx)

The cause of the discrepancy is that the FOL versions of a and e propositions lack "existential import." That is, x (Fx Gx) and x (Fx ?Gx) are both true when ?x Fx is true, that is, when there are no Fs. (These are the so-called "vacuously true" universal generalizations.) So `All Fs are Gs', on the modern reading, does not imply that there are Fs, and so does not imply that some Fs are Gs.

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c. The Presupposition Approach:

contraries

a

e

implies

implies

i

o

subcontraries

The simplest way to bring the ancient and modern squares into line is to assume that Aristotelian syllogistic is intended to be used only with non-empty predicates. If we assume that `F' and `G' are non-empty, we can keep the standard FOL translations of the four Aristotelian forms:

a All Fs are Gs e No Fs are Gs i Some Fs are Gs o Some Fs are not Gs

x (Fx Gx) x (Fx ?Gx) x (Fx Gx) x (Fx ?Gx)

Each of the four propositions is now taken to presuppose that the letters F, G, etc. are assigned non-empty sets.

Presupposition vs. Entailment

P entails Q P presupposes Q

If Q is false, P is false. If Q is false, P is neither true nor false.

The presupposition assumption, at a single stroke, disables all the counterexamples to the traditional entailment and contrariety relations. For the only way in which the FOL version of a (or e) could be true while i (or o) is false is if there are no Fs. Likewise, the only way in which a and e could both be true (or i and o both false) is if there are no Fs. So if the syllogistic propositions presuppose that there are Fs, there are no counterexamples.

This simple solution reinstates all the traditional relationships of contrariety, subcontrariety, and implication. But there are shortcomings with this approach, as well. For it is not enough to say that each simple predicate we use must be non-empty; we will also have to require that any logical operation we perform on predicates will also yield a non-empty result.

For example, `swan' is non-empty, and so is `black'. But suppose that, in fact, all swans are white. Still, we can conjoin the predicates `swan' and `black' to get `black swan'. This compound predicate is as a matter of fact empty. So we cannot allow it to be the subject of an a proposition, or else we will get embarrassing results. For if "all black swans have long necks" is true, then (since a implies i) so is "some black swans have long necks" and hence we are committed to the existence of black swans. But if "all black swans have long necks" is false, then (since a and o are contradictories) "some black swans don't have long necks" is true, and once again we are committed to the existence of black swans.

Thus we will have to say that syllogistic simply does not apply to such sentences. But the emptiness of `black swan' seems to be a factual matter. And should the applicability of an inferential system depend on such things? This seems to bring too much of the world into the logical system. We may only discover the emptiness of the

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